In the case of supervised IB theory [1]}, we can get the upper bound:
\(I_{\phi }(X;Z)-\beta I_{\theta }(Z;Y)\le & \mathbb {E}_{p_D(x)}[D_{KL}(q_{\phi }(z|x)\Vert p(z))]\nonumber \\&-\beta \mathbb {E}_{p(x,y)}[q_{\phi }(z|x)\log p_{\theta }(y|z)]\) 
